arXiv:cond-mat/9802220 · Physical Review Letters

Coulomb Drag of Luttinger Liquids and Quantum Hall Edges

Karsten Flensberg

Published Jul 6, 1998• 48 citations

Abstract

We study the transconductance for two coupled one-dimensional wires or edge states described by Luttinger liquid models. The wires are assumed to interact over a finite segment. We find for the interaction parameter $g\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/2$ that the drag rate is finite at zero temperature, which cannot occur in a Fermi-liquid system. The zero temperature drag is, however, cut off at low temperature due to the finite length of the wires. We also consider edge states in the fractional quantum Hall regime, and suggest that low temperature enhancement of the drag effect might be seen in this regime.

DOI: 10.1103/physrevlett.81.184

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arXiv:cond-mat/9802220 · Physical Review Letters

Coulomb Drag of Luttinger Liquids and Quantum Hall Edges

Karsten Flensberg

Published Jul 6, 1998• 48 citations

arXiv HTML PDF

Abstract

g\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/2

that the drag rate is finite at zero temperature, which cannot occur in a Fermi-liquid system. The zero temperature drag is, however, cut off at low temperature due to the finite length of the wires. We also consider edge states in the fractional quantum Hall regime, and suggest that low temperature enhancement of the drag effect might be seen in this regime.

DOI: 10.1103/physrevlett.81.184

Related papers

Coulomb drag between two spin-incoherent Luttinger liquids

Gregory A. Fiete, Karyn Le Hur, Leon Balents

In a one-dimensional electron gas at low enough density, the magnetic (spin) exchange energy $J$ between neighboring electrons is exponentially suppressed relative to the characteristic charge energy, the Fermi energy ${E}_{F}$ . At nonzero temperature $T$ , the energy hierarchy $J⪡T⪡{E}_{F}$ can be reached, and we refer to this as the spin-incoherent Luttinger liquid state. We discuss the Coulomb drag between two parallel quantum wires in the spin-incoherent regime, as well as the crossover to this state from the low-temperature regime by using a model of a fluctuating Wigner solid. As the temperature increases from zero to above $J$ for a fixed electron density, the $2{k}_{F}$ oscillations in the density-density correlations are lost. As a result, the temperature dependence of the Coulomb drag is dramatically altered and nonmonotonic dependence may result. Drag between wires of equal and unequal density are discussed, as well as the effects of weak disorder in the wires. We speculate that weak disorder may play an important role in extracting information about quantum wires in real drag experiments.In a one-dimensional electron gas at low enough density, the magnetic (spin) exchange energy $J$ between neighboring electrons is exponentially suppressed relative to the characteristic charge energy, the Fermi energy ${E}_{F}$. At nonzero temperature $T$, the energy hierarchy $J⪡T⪡{E}_{F}$ can be reached, and we refer to this as the spin-incoherent Luttinger liquid state. We discuss the Coulomb drag between two parallel quantum wires in the spin-incoherent regime, as well as the crossover to this state from the low-temperature regime by using a model of a fluctuating Wigner solid. As the temperature increases from zero to above $J$ for a fixed electron density, the $2{k}_{F}$ oscillations in the density-density correlations are lost. As a result, the temperature dependence of the Coulomb drag is dramatically altered and nonmonotonic dependence may result. Drag between wires of equal and unequal density are discussed, as well as the effects of weak disorder in the wires. We speculate that weak disorder may play an important role in extracting information about quantum wires in real drag experiments.

arXiv:cond-mat/0511715Physical Review BApr 5, 200660 citations

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D. N. Aristov

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Temperature Dependence of Coulomb Drag Between Finite-Length Quantum Wires

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We evaluate the Coulomb drag current in two finite-length Tomonaga-Luttinger-liquid wires coupled by an electrostatic backscattering interaction. The drag current in one wire shows oscillations as a function of the bias voltage applied to the other wire, reflecting interferences of the plasmon standing waves in the interacting wires. In agreement with this picture, the amplitude of the current oscillations is reduced with increasing temperature. This is a clear signature of non-Fermi-liquid physics because for coupled Fermi liquids the drag resistance is always expected to increase as the temperature is raised.We evaluate the Coulomb drag current in two finite-length Tomonaga-Luttinger-liquid wires coupled by an electrostatic backscattering interaction. The drag current in one wire shows oscillations as a function of the bias voltage applied to the other wire, reflecting interferences of the plasmon standing waves in the interacting wires. In agreement with this picture, the amplitude of the current oscillations is reduced with increasing temperature. This is a clear signature of non-Fermi-liquid physics because for coupled Fermi liquids the drag resistance is always expected to increase as the temperature is raised.

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Coulomb drag between quantum wires

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We study Coulomb drag in a pair of parallel one-dimensional electron systems within the framework of the Tomonaga-Luttinger model. We find that Coulomb coupling has a much stronger effect on one-dimensional wires than on two-dimensional layers: At zero temperature the transresistivity diverges, due to the formation of locked charge density waves. At temperature well above a crossover temperature ${T}^{*}$ the transresistivity follows a power law $\ensuremath{\rho}\ensuremath{\propto}{T}^{x},$ where the interaction-strength dependent exponent x is determined by the Luttinger liquid parameter ${K}_{c\ensuremath{-}}$ of the relative charge mode. At temperature below ${T}^{*}$ relative charge displacements are enabled by solitonic excitations, reflected by an exponential temperature dependence. The crossover temperature ${T}^{*}$ depends sensitively on the wire width, interwire distance, Fermi wavelength and the effective Bohr radius. For wire distances $\overline{d}\ensuremath{\gg}{k}_{F}^{\ensuremath{-}1}$ it is exponentially suppressed with ${T}^{*}{/E}_{F}\ensuremath{\sim}\mathrm{exp}[\ensuremath{-}\overline{d}{k}_{F}/(1\ensuremath{-}{K}_{c\ensuremath{-}})].$ The behavior changes drastically if each of the two wires develop spin gaps. In this case we find that the transresistivity vanishes at zero temperature. We discuss our results in view of possible experimental realizations in $\mathrm{GaAs}\ensuremath{-}{\mathrm{Al}}_{x}{\mathrm{Ga}}_{1\ensuremath{-}x}\mathrm{As}$ semiconductor structures.We study Coulomb drag in a pair of parallel one-dimensional electron systems within the framework of the Tomonaga-Luttinger model. We find that Coulomb coupling has a much stronger effect on one-dimensional wires than on two-dimensional layers: At zero temperature the transresistivity diverges, due to the formation of locked charge density waves. At temperature well above a crossover temperature ${T}^{*}$ the transresistivity follows a power law $\ensuremath{\rho}\ensuremath{\propto}{T}^{x},$ where the interaction-strength dependent exponent x is determined by the Luttinger liquid parameter ${K}_{c\ensuremath{-}}$ of the relative charge mode. At temperature below ${T}^{*}$ relative charge displacements are enabled by solitonic excitations, reflected by an exponential temperature dependence. The crossover temperature ${T}^{*}$ depends sensitively on the wire width, interwire distance, Fermi wavelength and the effective Bohr radius. For wire distances $\overline{d}\ensuremath{\gg}{k}_{F}^{\ensuremath{-}1}$ it is exponentially suppressed with ${T}^{*}{/E}_{F}\ensuremath{\sim}\mathrm{exp}[\ensuremath{-}\overline{d}{k}_{F}/(1\ensuremath{-}{K}_{c\ensuremath{-}})].$ The behavior changes drastically if each of the two wires develop spin gaps. In this case we find that the transresistivity vanishes at zero temperature. We discuss our results in view of possible experimental realizations in $\mathrm{GaAs}\ensuremath{-}{\mathrm{Al}}_{x}{\mathrm{Ga}}_{1\ensuremath{-}x}\mathrm{As}$ semiconductor structures.

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Coulomb drag effect between Luttinger liquids

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The exact solution of the one-dimensional Hubbard model, the supersymmetric $t\ensuremath{-}J$ model, and the gas of fermions interacting via a $\ensuremath{\delta}$-function potential is used to calculate the drag current between two parallel quantum wires. The critical exponent for the drag current at low temperatures is obtained by means of the mesoscopic energy spectrum and conformal field theory. For a repulsive interaction between the carriers the drag current is opposite to the driving current, while for attractive potentials the two currents are parallel.The exact solution of the one-dimensional Hubbard model, the supersymmetric $t\ensuremath{-}J$ model, and the gas of fermions interacting via a $\ensuremath{\delta}$-function potential is used to calculate the drag current between two parallel quantum wires. The critical exponent for the drag current at low temperatures is obtained by means of the mesoscopic energy spectrum and conformal field theory. For a repulsive interaction between the carriers the drag current is opposite to the driving current, while for attractive potentials the two currents are parallel.

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Transport in a spin-incoherent Luttinger liquid

Gregory A. Fiete, Karyn Le Hur, Leon Balents

We theoretically investigate transport in a spin-incoherent one-dimensional electron system, which may be realized in quantum wires at low-electron density and finite temperature. Both the pure and disordered cases are considered, both in finite wires and in the thermodynamic limit. The effect of Fermi-liquid leads attached to the finite-length system is also addressed. In the infinite system, we find a phase diagram identical to that obtained for a spinless Luttinger liquid, provided we make the identification $g=2{g}_{c}$ , where $g$ is the interaction parameter in a spinless Luttinger liquid and ${g}_{c}$ is the interaction parameter of the charge sector of a Luttinger-liquid theory for electrons with spin. For a finite-length wire attached to Fermi-liquid leads, the transport depends on the details of the disorder in the wire. A simple picture for the crossover from the spin-incoherent regime to the spin-coherent regime as the temperature is varied is also discussed, as well as some physical implications.We theoretically investigate transport in a spin-incoherent one-dimensional electron system, which may be realized in quantum wires at low-electron density and finite temperature. Both the pure and disordered cases are considered, both in finite wires and in the thermodynamic limit. The effect of Fermi-liquid leads attached to the finite-length system is also addressed. In the infinite system, we find a phase diagram identical to that obtained for a spinless Luttinger liquid, provided we make the identification $g=2{g}_{c}$, where $g$ is the interaction parameter in a spinless Luttinger liquid and ${g}_{c}$ is the interaction parameter of the charge sector of a Luttinger-liquid theory for electrons with spin. For a finite-length wire attached to Fermi-liquid leads, the transport depends on the details of the disorder in the wire. A simple picture for the crossover from the spin-incoherent regime to the spin-coherent regime as the temperature is varied is also discussed, as well as some physical implications.

arXiv:cond-mat/0505186Physical Review BSep 13, 200533 citations

g\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/2

In a one-dimensional electron gas at low enough density, the magnetic (spin) exchange energy

J

between neighboring electrons is exponentially suppressed relative to the characteristic charge energy, the Fermi energy

{E}_{F}

. At nonzero temperature

T

, the energy hierarchy

J⪡T⪡{E}_{F}

can be reached, and we refer to this as the spin-incoherent Luttinger liquid state. We discuss the Coulomb drag between two parallel quantum wires in the spin-incoherent regime, as well as the crossover to this state from the low-temperature regime by using a model of a fluctuating Wigner solid. As the temperature increases from zero to above

J

for a fixed electron density, the

2{k}_{F}

oscillations in the density-density correlations are lost. As a result, the temperature dependence of the Coulomb drag is dramatically altered and nonmonotonic dependence may result. Drag between wires of equal and unequal density are discussed, as well as the effects of weak disorder in the wires. We speculate that weak disorder may play an important role in extracting information about quantum wires in real drag experiments.

T

. The most important application of our approach is the analysis of the Coulomb drag by small momentum transfer between two LLs, which is only possible due to curvature. Analyzing the ac transconductivity in the one-dimensional drag setup, we confirm the results by Pustilnik et al. for

T

dependence of drag resistivity, ${R}_{12}\ensuremath{\sim}{T}^{2}$ at high and ${R}_{12}\ensuremath{\sim}{T}^{5}$ at low temperatures. The bosonization allows for treating both intra- and interwire electron-electron interactions in all orders, and we calculate exact prefactors in the low-

T

drag regime. The crossover temperature between the two regimes is ${T}_{1}\ensuremath{\sim}{E}_{F}\ensuremath{\Delta}$, with $\ensuremath{\Delta}$ relative difference in plasmon velocities. We show that $\ensuremath{\Delta}\ensuremath{\ne}0$ even for identical wires, due to lifting of degeneracy by interwire interaction,

{U}_{12}

, leading to crossover from ${R}_{12}\ensuremath{\sim}{U}_{12}^{2}{T}^{2}$ to ${R}_{12}\ensuremath{\sim}{T}^{5}∕{U}_{12}$ at $T\ensuremath{\sim}{U}_{12}$.

{T}^{*}

the transresistivity follows a power law $\ensuremath{\rho}\ensuremath{\propto}{T}^{x},$ where the interaction-strength dependent exponent x is determined by the Luttinger liquid parameter ${K}_{c\ensuremath{-}}$ of the relative charge mode. At temperature below

{T}^{*}

relative charge displacements are enabled by solitonic excitations, reflected by an exponential temperature dependence. The crossover temperature

{T}^{*}

depends sensitively on the wire width, interwire distance, Fermi wavelength and the effective Bohr radius. For wire distances $\overline{d}\ensuremath{\gg}{k}_{F}^{\ensuremath{-}1}$ it is exponentially suppressed with ${T}^{*}{/E}_{F}\ensuremath{\sim}\mathrm{exp}[\ensuremath{-}\overline{d}{k}_{F}/(1\ensuremath{-}{K}_{c\ensuremath{-}})].$ The behavior changes drastically if each of the two wires develop spin gaps. In this case we find that the transresistivity vanishes at zero temperature. We discuss our results in view of possible experimental realizations in $\mathrm{GaAs}\ensuremath{-}{\mathrm{Al}}_{x}{\mathrm{Ga}}_{1\ensuremath{-}x}\mathrm{As}$ semiconductor structures.

g=2{g}_{c}

, where

g

is the interaction parameter in a spinless Luttinger liquid and

{g}_{c}

is the interaction parameter of the charge sector of a Luttinger-liquid theory for electrons with spin. For a finite-length wire attached to Fermi-liquid leads, the transport depends on the details of the disorder in the wire. A simple picture for the crossover from the spin-incoherent regime to the spin-coherent regime as the temperature is varied is also discussed, as well as some physical implications.